Question

Use the following expression to answer the questions below:

Answer 1

We want to simplify:

$(\frac{1}{x+3}+ \frac{6}{ x^{2} +4x+3})* \frac{x+3}{x+1}$

first we factorize $x^{2} +4x+3$ as $(x+3)(x+1)$

then multiply $\frac{1}{x+3}$ by $\frac{x+1}{x+1}$ to equalize the denominators of the first expression, and add the fractions.

$(\frac{1}{x+3}*\frac{x+1}{x+1} + \frac{6}{(x+3)(x+1)})* \frac{x+3}{x+1}$
$\frac{x+7}{(x+3)(x+1)}* \frac{x+3}{x+1}$

simplify x+3:

$\frac{x+7}{(x+1)}* \frac{1}{x+1}= \frac{x+7}{ x^{2} +2x+1}$

a. 
$\frac{x+7}{ x^{2} +2x+1}$

b. 
steps already described

c. 
x cannot be -1, nor -3 because the fractions with denominators (x+1) and (x+3) would not be defined.